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1st Edition, Fourth PrintingAbraham Weishaus, Ph.D., F.S.A., C.F.A., M.A.A.A.

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Contents

1 Introduction to Derivatives 1Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Project Analysis 52.1 NPV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Project analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Break-even analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.3 Scenario analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Risk measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.1 Four risk measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Coherent risk measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Monte Carlo Simulation 19Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Efficient Markets Hypothesis (EMH) 51Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5 Mean-Variance Portfolio Theory 57Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6 Capital Asset Pricing Model (CAPM) 756.1 Required return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.2 CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7 Cost of Capital 877.1 Estimating CAPM’s parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.2 Project cost of capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88


8 Behavioral Finance and Multifactor Models 1018.1 Behavioral finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018.2 Multifactor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102


9 Capital Structure 107Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

IFM Study Manual—1st edition 4th printingCopyright ©2019 ASM

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iv CONTENTS

10 The Effect of Taxes on Capital Structure 117Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

11 Other Factors Affecting Optimal Debt-Equity Ratio 12511.1 Bankruptcy and Financial Distress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12511.2 Agency costs and benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12711.3 Asymmetric information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128


12 Equity Financing 13512.1 Sources of capital for private companies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13512.2 Allocation of company value among investors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13512.3 Going public: the IPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13712.4 IPO puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138


13 Debt Financing 14313.1 Corporate Debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14313.2 Other Debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144


14 Forwards 14714.1 How a forward works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14714.2 Pricing a Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14814.3 Additional Comments on Forwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

14.3.1 Forward Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15114.3.2 Comparison of Forward Price to Expected Future Price . . . . . . . . . . . . . . . . . . . . . . 151

14.4 Synthetic forwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

15 Variations on the Forward Concept 16115.1 Prepaid forwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16115.2 Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162


16 Options 16716.1 Call options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16716.2 Put options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168


17 Option Strategies 17717.1 Options with Underlying Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17717.2 Synthetic Forwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17917.3 Bear Spreads, Bull Spreads, and Collars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

17.3.1 Spreads: buying an option and selling another option of the same kind . . . . . . . . . . . . . 18017.3.2 Collars: buying one option and selling an option of the other kind . . . . . . . . . . . . . . . 183

17.4 Straddles, strangles, and butterfly spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185


CONTENTS v


18 Put-Call Parity 20118.1 Stock put-call parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20218.2 Synthetic stocks and Treasuries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20318.3 Synthetic options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20418.4 Currency options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204


19 Comparing Options 22119.1 Bounds for Option Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22119.2 Early exercise of American options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22319.3 Time to expiry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22619.4 Different strike prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

19.4.1 Three inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22919.4.2 Options in the money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

20 Binomial Trees—Stock, One Period 24520.1 Risk-neutral pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24520.2 Replicating portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24720.3 Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251


21 Binomial Trees—General 27121.1 Multi-period binomial trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27121.2 American options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27221.3 Currency options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27421.4 Options on Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27421.5 Other assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278


22 Binomial Trees: Understanding Early Exercise of Options 303Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

23 Modeling Stock Prices with the Lognormal Distribution 30723.1 The normal and lognormal distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

23.1.1 The normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30723.1.2 The lognormal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30823.1.3 Jensen’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

23.2 The lognormal distribution as a model for stock prices . . . . . . . . . . . . . . . . . . . . . . . . . . . 31023.2.1 Stocks without dividends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31023.2.2 Stocks with dividends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31223.2.3 Prediction intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

23.3 Conditional payoffs using the lognormal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322


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24 The Black-Scholes Formula 32724.1 Black-Scholes Formula for common stock options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32824.2 Black-Scholes formula for currency options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33024.3 Black-Scholes formula for options on futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331


25 The Black-Scholes Formula: Greeks 34725.1 Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

25.1.1 Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34825.1.2 Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35325.1.3 Vega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35325.1.4 Theta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35525.1.5 Rho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35925.1.6 Psi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36225.1.7 Greek measures for portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

25.2 Elasticity and related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36725.2.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36725.2.2 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36825.2.3 Elasticity of a portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

25.3 What will I be tested on? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

26 Delta Hedging 38526.1 Overnight profit on a delta-hedged portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38526.2 The delta-gamma-theta approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38926.3 Hedging multiple Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39126.4 What will I be tested on? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392


27 Asian, Barrier, and Compound Options 41127.1 Asian options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41127.2 Barrier options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41527.3 Maxima and minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41827.4 Compound options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

27.4.1 Compound option parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

28 Gap, Exchange, and Other Options 43528.1 Gap options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

28.1.1 Definition of gap options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43528.1.2 Pricing gap options using Black-Scholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43628.1.3 Delta hedging gap options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

28.2 Exchange options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44028.3 Other exotic options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

28.3.1 Chooser options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44128.3.2 Forward start options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453


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29 Supplementary Questions—Derivatives 461Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

30 Real Options 47730.1 Decision trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47730.2 Black-Scholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47930.3 Option to abandon project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480


31 Actuarial Applications of Options 48731.1 Variable annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48731.2 Mortgage guaranty insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48931.3 Other insurance features that constitute options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48931.4 Static hedging strategies using exotic options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49031.5 Dynamic hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49131.6 Hedging catastrophe risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491


Practice Exams 495

1 Practice Exam 1 497











Appendices 591

A Solutions for the Practice Exams 593Solutions for Practice Exam 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593Solutions for Practice Exam 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599Solutions for Practice Exam 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605Solutions for Practice Exam 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612Solutions for Practice Exam 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619Solutions for Practice Exam 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626


viii CONTENTS

Solutions for Practice Exam 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634Solutions for Practice Exam 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642Solutions for Practice Exam 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650Solutions for Practice Exam 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658Solutions for Practice Exam 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668

B Solutions to Old Exams 677B.1 Solutions to SOA Exam MFE, Spring 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677B.2 Solutions to CAS Exam 3, Spring 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681B.3 Solutions to CAS Exam 3, Fall 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684B.4 Solutions to Exam MFE/3F, Spring 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688B.5 Solutions to Advanced Derivatives Sample Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 691

C Lessons Corresponding to Questions on Released and Practice Exams 701

D Standard Normal Distribution Function Table 705


Lesson 2

Project Analysis

Reading: Corporate Finance 8.5, IFM 21-18 2–3

This lesson begins the corporate finance part of the course. From here up to and including Lesson 13, when nottold otherwise, assume that interest rates are annual effective.

2.1 NPV

When a company considers embarking on a project, it must verify that this project will meet the company’s financialgoals. The measure we will use for this is NPV, or net present value. To compute the NPV we calculate the freecash flows of the project. The free cash flows are the cash amounts generated by the project itself, both positive andnegative, year by year. Cash flows do not include non-cash accounting items, such as depreciation.1 The free cashflows also do not include cash flows from financing used to support the project. If a loan is taken to pay the project’sinitial expenses, neither the loan nor interest on the loan is part of free cash flows. Free cash flows are purely cashgenerated by the project itself.

The NPV is the present value, at the start of the project, of the project’s free cash flows. At what interest rate isthe NPV calculated? Usually the NPV is calculated at the interest rate the company must pay to finance the project.In other words, the NPV is calculated at the interest rate that has to be paid to investors in order to get them to investin the project. This interest rate is called the cost of capital. We will discuss what the cost of capital should be in thefollowing lessons.Example 2A A life insurance company is considering developing a new Universal Life product. It will cost $5million, payable immediately, to develop this product, and developing the product will take a year. The companyestimates free cash flows will be $−1 million at the end of the first year, followed by $1 million per year for 5 years,and will then decrease at a compounded rate of 10% per year after that.

The company’s cost of capital is 12%.Calculate the NPV of the project.

Solution: The following time line shows the cash flows in millions:

0 1 2 3 4 5 6 7 8

$−5 $−1 $1 $1 $1 $1 $1 $0.9 $0.81

The NPV generated during the first 6 years is

−5,000,000 − 1,000,0001.12 + 1,000,000

(a5 0.121.12

)� −5,892,857 + 1,000,000

(1 − 1/1.125

0.12(1.12))� −2,674,307

After 6 years, free cash flows form a geometric series with first term 900,000/1.127 and ratio 0.9/1.12. The NPVgenerated after year 6, in millions, is

900,000(

1/1.127

1 − 0.9/1.12

)� 2,072,582

Total NPV is −2,674,307 + 2,027,582 � −601,725 . �

1However, for insurance products, they include changes in reserves. A company must set aside cash to support the reserves, although thiscash may be invested.


5

6 2. PROJECT ANALYSIS

Companies should invest in a project only if NPV > 0. Otherwise they destroy the value of the company.If we assume that free cash flows are constant, they form a perpetuity. As you learned in Financial Mathematics,

the present value of an immediate perpetuity of 1 per year is 1/i, where i is the interest rate. If the free cash flowsare 1 in the first year and grow at compounded rate g, then their present value is

NPV �

∞∑n�1

(1 + g)n−1

(1 + i)n

�1/(1 + i)

1 − (1 + g)/(1 + i)�

1/(1 + i)(i − g)/(1 + i) �

1i − g

(2.1)

?Quiz 2-1 A company is considering a project. This project will require an investment of 10million immediatelyand will generate free cash flows of 1 million per year at the end of one year, increasing at a compounded rateof 3% per year perpetually.

The cost of capital is 9%.Calculate the NPV of the project.

2.2 Project analysis

2.2.1 Break-even analysisCompanies analyze the risk in a project. One way to analyze the risk is to vary the assumptions used to calculatethe NPV with the changed assumptions. Break-even analysis consists of determining the value of each assumptionparameter for which the NPV is 0, assuming that the other assumption parameters are at their baseline values.

Calculation of IRR is an example of break-even analysis. IRR, the internal rate of return, is an alternative profitmeasure to NPV. The IRR is the interest rate r such that the present value of free cash flows at r is 0. Assuming theusual pattern of negative free cash flows initially followed by positive free cash flows, IRR is the highest interest ratefor which the NPV is at least 0. Thus IRR is the highest interest rate for which the company breaks even.

A similar analysis can be done for the other parameters. A break-even analysis calculates the break-even level ofnumber of sales, expenses, sales price, level of cash flows per year, and any other parameter.Example 2B A project requires an immediate investment of 19 million. It is expected to generate free cash flows of2 million per year at the end of the first year, growing 2% per year perpetually. The cost of capital is 12%.

Perform a break-even analysis on the rate of growth of free cash flows.

Solution: Let g be the growth rate. We want to solve −19 +2

0.12−g � 0

−19 +2

0.12 − g� 0

20.12 − g

� 19

0.12 − g �219

g � 0.01474 �

?Quiz 2-2 A project to develop a new product requires an immediate investment of 9 million. It will thengenerate free cash flows of 1 million per year starting with the end of the first year, until the product becomesobsolete and cannot be sold. The cost of capital is 10%.

Perform a break-even analysis on the number of years the product must sell.


2.3. RISK MEASURES 7

2.2.2 Sensitivity analysis

Sensitivity analysis consists of calculating the change in the NPV resulting from a change in a parameter. Typicallyone sets the parameter to its value in the worst possible case and the best possible case, and calculates the NPV forboth cases. This analysis shows which parameters have the greatest impact on the NPV.Example 2C A project to develop a new product requires an immediate investment of 15 million. Free cash flowsgenerated by this project are 20% of sales. Sales are expected to be level, and to continue for a certain number ofyears, at which point the product becomes obsolete. The best and worst cases for each assumption are:

Worst case Baseline Best caseAnnual sales ($ million) 20 25 30Number of years 3 5 7Cost of capital 0.16 0.12 0.08

Perform a sensitivity analysis on the three factors listed in the table. Which factor is the NPV most sensitive to?

Solution: We’ll do all calculations in millions.For annual sales s:

NPV � −15 + 0.2sa5 0.12 � −15 + 0.2s(1 − 1/1.125

0.12

)

which is −$0.581 million for s � 20 and $6.629 million for s � 30, a variation of $7.210 millionFor number of years n:

NPV � −15 + 5an 0.12 � −15 + 5(1 − 1/1.12n

0.12

)

which is −$2.991 million for n � 3 and $7.818 million for n � 7, a variation of $10.809 million.For cost of capital r:

NPV � −15 + 5a5 r � −15 + 5(1 − 1/(1 + r)5

r

)

which is $1.371 million for r � 0.16 and $4.964 for r � 0.08, a variation of $3.593 million.We see that number of years of sales is the assumption to which NPV is most sensitive. �

2.2.3 Scenario analysis

Often parameters are correlated and should not be analyzed separately. For example, increasing the price of aproduct may lower sales. Scenario analysis consists of calculating the NPV for various scenarios. A scenario mayvary two parameters in a consistent manner, leaving the other parameters unchanged if they are uncorrelated.

2.3 Risk measures

In the previous section we analyzed risk by varying parameters. An alternative method for analyzing risk is toassign a number to the project indicating its riskiness. This section discusses such risk measures. Each of these riskmeasures is a function from a random variable to a real number. The random variable may be profits, returns oninvestment, or aggregate loss amounts paid by an insurance company. Notice that the direction of risk for aggregateloss amounts is the opposite of profits or returns: the risk is that profits or returns are low and that loss amountsare high.

2.3.1 Four risk measures

We will discuss four risk measures: variance, semi-variance, VaR, and TVaR.



Variance

Variance is a popular risk measure and will be used in mean-variance portfolio theory, which we discuss starting inLesson 5. If R is the random variable for the return on an investment, the mean return is µ and the variance is

Var(R) � σ2� E[(R − µ)2] � E[R2] − µ2 (2.2)

An equivalent riskmeasure is the square root of the variance, or the standard deviation σ. We’ll also use the notationSD(R) for the standard deviation.

SD(R) �√

Var(R) � σThe standard deviation of the rate of return is also called the volatility of the rate of return.

The variance may be estimated from a sample using the formula

σ̂2�

n∑i�1

(Ri − R̄)2n

where R̄ is the sample mean. That is the formula given in the study note, but usually the denominator is n − 1instead of n to make this estimate unbiased. In fact, the formula for estimating volatility given in the Berk/DeMarzotextbook is equation (5.1) on page 57, and that formula divides by n − 1.

Semi-variance

Since we are more concerned with underperformance than overperformance, at least for profits and rates of returnon investments, we may prefer the downside semi-variance, which we’ll refer to as the semi-variance for short, asa measure of risk. The semi-variance considers the square difference from the mean only when that difference isnegative. It is defined by

σ2SV � E

[min

(0, (R − µ))2] (2.3)

The semi-variance is positive even though it is based on negative differences from the mean, since the differencesare squared. The square root of the downside semi-variance is the downside standard deviation.Example 2D The random variable X has an exponential distribution with mean 1:

fX(x) � e−x , x > 0

Calculate the semi-variance of X.

Solution: We integrate min(0, x − 1)2 over the density function. This minimum is 0 for x > 1, so we only integrateup to 1. We’ll integrate by parts twice.

∫ 1

0(x − 1)2e−x dx � −(x − 1)2e−x

��10+ 2

∫ 1

0(x − 1)e−x dx

� 1 + 2(−(x − 1)e−x

��10+

∫ 1

0e−x dx

)

� 1 − 2 + 2(1 − e−1) � 1 − 2e−1� 0.264241 �

The sum of the downside semi-variance and the upside semi-variance is the variance:

E[(X − µ)2] �∫ ∞

−∞(x − µ)2 fX(x)dx

�

∫ µ

−∞(x − µ)2 fX(x)dx +

∫ ∞

µ(x − µ)2 fX(x)dx

The first term of the last expression is the downside semi-variance and the second term is the upside semi-variance.


2.3. RISK MEASURES 9

The semi-variance may be estimated from a sample by

σ̂2SV �

1n

n∑i�1

min(0, (Ri − R̄))2 (2.4)

?Quiz 2-3 You are given the following sample:

5 10 15 20 25

Calculate the sample semi-variance.

Value-at-Risk (VaR)

The VaR of a random variable X at level α is the 100α percentile of the random variable. For a continuous randomvariable, it is x such that Pr(X ≤ x) � α. For profits or rates of return, where the risk is that X is low, α is picked low,with values like 0.05, 0.025, 0.01, 0.005. For aggregate insurance losses, where the risk is that X is high, α is pickedhigh, with values like 0.95, 0.975, 0.99, 0.995.

The VaR is calculated by inverting the cumulative distribution function:

VaRα(X) � F−1X (α) (2.5)

Example 2E Profits have a distribution with the following density function:

f (x) � 3(1 + x)4 x > 0

Calculate VaR of profits at the 0.01 level.

Solution: Integrate f (x) to obtain the cumulative distribution function, then invert that function at 0.01.

F(x) �∫ x

0

3 dt(1 + t)4 � 1 − 1

(1 + x)31 − 1(1 + x)3 � 0.01

1 + x �3√

1/0.99 � 1.003356

x � 0.003356 �

To estimate VaR from a sample, the sample is ordered from lowest to highest, and then the 100α percentile isselected. This percentile is not well-defined since a sample is a discrete distribution, so some rule for selecting thepercentile is needed. For example, if the sample is size 1000 and α � 0.05, then one might set the sample VaR equalto the 50th order statistic (most conservative), the 51st order statistic, or some weighted average of the two, such asthe smoothed empirical percentile defined in the Exam STAM syllabus.

Tail value-at-risk

While value-at-risk identifies the amount which returns or profits will exceed a great proportion (1−α to be exact) ofthe time, it doesn’t consider the severity of the downside risk in the remaining α of the time. Tail value-at-risk, alsoknown as Conditional Tail Expectation (CTE) or Expected Shortfall measures this risk. It is defined as the expectedvalue of the random variable given that it is below the 100α percentile for downside risk

TVaRα(X) � E[X | X < VaRα(X)] �∫ VaRα(X)−∞ x f (x)dx

α(2.6)



or the expected value of the random variable given that it is above the 100α percentile for upside risk, like aggregatelosses

TVaRα(X) � E[X | X > VaRα(X)] �∫ ∞

VaRα(X) x f (x)dx

1 − α (2.7)

You should be able to figure out whether upside or downside risk is present based on what is being analyzed, butif not, if α < 0.5, presumably the risk is downside and if α > 0.5, presumably the risk is upside.

It may be difficult or impossible to evaluate the intergral needed to calculate TVaR.TVaR can be estimated from a sample. Select the bottom or top α proportion of the items of the sample and

calculate their mean. For example, if the sample is size 1000 and α � 0.05, average the lowest 50 items of the sampleto calculate downside risk.

2.3.2 Coherent risk measuresLet’s list four desirable properties of a risk measure g(X).

1. Translation invariance. Adding a constant to the random variable should add the same constant to the riskmeasure. Or:

g(X + c) � g(X) + c

This is reasonable, since a constant gain or loss generates no risk beyond its amount.

2. Positive homogeneity. Multiplying the random variable by a positive constant should multiply the riskmeasure by the same constant:

g(cX) � c g(X)This is reasonable, since expressing the random variable in a different currency (for example) should not affectthe risk measure.

3. Subadditivity. For any two random losses X and Y, the risk measure for X + Y should not be greater than thesum of the risk measures for X and Y separately:

g(X + Y) ≤ g(X) + g(Y)This is reasonable, since combining losses may result in diversification and reducing the total risk measure,but it should not be possible by breaking a risk into two sub-risks to reduce the total risk measure.This is formeasuringupside risk. Formeasuringdownside risk, the subadditivity property becomes g(X+Y) ≥g(X) + g(Y).2

4. Monotonicity. For any two random losses X and Y, if X is always less than Y, or even if the probability thatX is less than or equal to Y is 1, then the risk measure for X should be no greater than the risk measure for Y.

g(X) ≤ g(Y) if Pr(X ≤ Y) � 1

This is reasonable, since X clearly has no more risk than Y.This is for measuring upside risk. For measuring downside risk, the monotonicity property becomes g(X) ≥g(Y) if Pr(X ≥ Y) � 1.2

Risk measures satisfying all four of these properties are called coherent.Risk measures with variance in their formula (such as variance itself and semi-variance) fail the monotonicity

property, since a constant has less variance than a random variable that varies, even if the random variable is alwaysless than the constant.

Value-at-risk is not subadditive and therefore not coherent, but tail value-at-risk is coherent. Value-at-risk satisfiesthe other properties. In special cases, such as when all distributions under consideration are normal, value-at-riskis coherent.

2The study note does not mention that the inequalities for coherence for downside risks are reversed.


EXERCISES FOR LESSON 2 11

Table 2.1: Formula Summary

NPV

NPV �

∞∑n�0

FCFn

(1 + r)n

whereFCFn is free cash flow at time nr is the cost of capital

If free cash flows are k at time 1 and grow at constant rate g, and the cost of capital is r, then their NPV is

kr − g

(2.1)

Downside semi-variance:σ2

SV � E[min

(0, (R − µ))2] (2.3)

Sample downside semi-variance:

σ̂2SV �

1n

n∑i�1

min(0, (Ri − R̄))2 (2.4)

Value-at-risk:VaRα(X) � F−1

X (α) (2.5)

TVaR for downside risk:

TVaRα(X) � E[X | X < VaRα(X)] �∫ VaRα(X)−∞ x f (x)dx

α(2.6)

TVaR for upside risk:

TVaRα(X) � E[X | X > VaRα(X)] �∫ ∞

VaRα(X) x f (x)dx

1 − α (2.7)

Exercises

2.1. A project requires an immediate investment of 12 million and an additional investment of 1 million per yearfor 5 years starting at the end of year 1. The project will generate free cash flows (ignoring the investment cash flows)of 1.5 million in year 1, growing 2% per year perpetually. The cost of capital is 10%.

Calculate the NPV of this project.

2.2. A project to produce new widgets requires a $10 million investment paid immediately. Installing the ma-chinery will take one year, during which time no widgets will be sold. It is expected that the sale of widgets willgenerate $2 million of free cash flows in year 2, growing $200,000 per year until year 11, at which time they willbecome obsolete and will not be sold any more.

The cost of capital is 10%.Calculate the NPV of this project.


Exercises continue on the next page . . .


2.3. A project to produce desks requires an investment of $20 million immediately. The machinery will last for 7years, at which point the project ends. You are given:

(i) The desks will sell for $500 apiece.(ii) The same number of desks will be sold each year.(iii) There will be fixed costs of $1 million per year, and the variable costs associated with manufacturing and

selling the desks are $200 apiece.(iv) The revenues from selling the desks and the associated fixed and variable costs occur at the end of each

year.(v) The cost of capital is 12%.

Based on a break-even analysis, calculate the number of desks per year that must be sold.

2.4. A project requires an immediate investment of 8 million. An additional investment of 2 million is required atthe end of year 1. Starting in the second year, the project will generate free cash flows of 1 million per year, growing3% per year perpetually.

Based on a break-even analysis, determine the cost of capital to break even.

2.5. A project requires an investment of 8 million. The following are baseline, best case, and worst case assump-tions:

Worst case Baseline Best caseFree cash flows in first year 1.1 1.2 1.3Rate of growth of free cash flows 0 0.03 0.05Number of years of free cash flows 7 10 13

The cost of capital is 0.10.Which of the three assumptions in the table is the NPV most sensitive to?

2.6. A company invests 8 million in a project to produce a new product. The product can be perpetually. Asensitivity analysis considers the following assumptions:

Worst case Baseline Best caseAnnual number of units sold 1,000,000 1,200,000 1,500,000Price per unit 1.25 1.50 1.60Expenses, as percentage of sales price 23% 20% 15%

The cost of capital is 0.15.To which assumption is the NPV most sensitive?

2.7. You are given the following sample:

1 3 7 15 25 39

Calculate the downside semi-variance.

2.8. A random variable X follows a normal distribution with µ � 20, σ2 � 100.Calculate the downside standard deviation of X.


Exercises continue on the next page . . .

EXERCISES FOR LESSON 2 13

2.9. A random variable X has the following probability density function:

f (x) �{

2x 0 ≤ x ≤ 10 otherwise

Calculate the downside semi-variance of X.

2.10. Profits X have the following cumulative distribution function:

F(x) � e−1000/x x > 0

Calculate the value-at-risk at 1%.


F(x) �

1 −(1000

x

)2

x > 1000

0 otherwise

Calculate the value-at-risk at 0.5%.

2.12. Losses on an insurance are distributed as follows:

Greater than Less than or equal to Probability0 1000 0.451000 2000 0.252000 5000 0.225000 10000 0.0510000 20000 0.03

Within each range losses are uniformly distributed.Calculate the tail value-at-risk for losses at 95%.


F(x) � 1 − e−x/1000 x > 0

Calculate the tail value-at-risk at 5%.

2.14. For a simulation with 100 runs, the largest 20 values are920 920 922 925 926 932 939 940 943 945948 952 959 962 969 976 989 1005 1032 1050

Estimate TVaR at 95% from this sample.

2.15. Consider the risk measure g(X) � E[X2]. Assume it is used only for nonnegative random variables.Which coherence properties does it satisfy?

2.16. Consider the risk measure g(X) � E[√X]. Assume it is used only for nonnegative random variables.Which coherence properties does it satisfy?

Finance and Investment sample questions: 27,34,35,42



Solutions

2.1. The present value of the investment is 12 + (1 − 1/1.15)/0.1 � 15.791 million. The present value of the freecash flows is 1.5/(0.1 − 0.02) � 18.75 million. The NPV is 18.75 − 15.791 � 2.959 million .2.2. At time 1, the present value of the cash flows from the widgets is 1,800,000a10 + 200,000(Ia)10 .

a10 �1 − 1/1.110

0.1 � 6.144567

Üa10 � (6.144567)(1.1) � 6.759024

(Ia)10 �6.759024 − 10/1.110

0.1 � 29.03591

So the present value of the cash flows at time 1 is 6.144567(1,800,000)+ 29.03591(200,000) � 16,867,404. Discountingto time 0 and subtracting the investment, the NPV is 16,867,403/1.1 − 10,000,000 � $5,334,002 .2.3. Present value of investment and fixed expenses is

20 + a7 � 20 +1 − 1/1.127

0.12 � 24.563757 million

Present value of net profit from sale of 1 desk per year is

300(1 − 1/1.127

0.12

)� 1369.127

So to break even, 24,563,757/1369.127 � 17,941 desks per year must be sold.2.4. Let r be the cost of capital. At time 1, the present value of future free cash flows is 1/(r − 0.03) in millions.Thus at time 0 the present value of these cash flows is 1

/ ((1 + r)(r − 0.03)) . We want r such that

−8 − 21 + r

+1

(1 + r)(r − 0.03) � 0

−8(1 + r)(r − 0.03) − 2(r − 0.03) + 1 � 08r2

+ 9.76r − 1.3 � 0

r � 0.121163

2.5. The present value at annual effective rate r of cash flows for n years at the end of each year starting at 1 andgrowing at a rate of g is

11 + r

n−1∑k�0

(1 + g1 + r

) k

�1

1 + r1 − (1+g

1+r

)n

1 − (1 + g)/(1 + r) �1 − (1+g

1+r

)n

r − g

In the following, all numbers are in millions. For the free cash flows in first year assumption, the NPVs of the worstand best cases are:

−8 + 1.1(1 − (1.03/1.1)10

0.10 − 0.03

)� −0.42788

−8 + 1.3(1 − (1.03/1.1)10

0.10 − 0.03

)� 0.94887

with difference 1.37675.


EXERCISE SOLUTIONS FOR LESSON 2 15

For the rate of growth assumption, the NPVs of the worst and best cases are:

−8 + 1.2(1 − (1/1.1)10

0.10

)� −0.62652

−8 + 1.2(1 − (1.05/1.1)10

0.10 − 0.05

)� 0.92777

with difference 1.55429For the number of years of free cash flows assumption, the NPVs of the worst and best cases are:

−8 + 1.2(1 − (1.03/1.1)7

0.10 − 0.03

)� −1.67634

−8 + 1.2(1 − (1.03/1.1)13

0.10 − 0.03

)� 1.85060

with difference 3.52694.The most sensitive assumption is number of years .

2.6. We will ignore the investment cost, which is the same in all scenarios.For annual units sold, the NPVs of the worst and best cases are:

1,000,000(1.50)(0.8)0.15 � 8,000,000

1,500,000(1.50)(0.8)0.15 � 12,000,000

with difference 4,000,000.For price per unit, the NPVs of the worst and best cases are:

1,200,000(1.25)(0.8)0.15 � 8,000,000

1,200,000(1.60)(0.8)0.15 � 10,240,000

with difference 2,240,000.For expenses, the NPVs of the worst and best cases are:

1,200,000(1.50)(0.77)0.15 � 9,240,000

1,200,000(1.50)(0.85)0.15 � 10,200,000

with difference 960,000.Annual units sold has the highest sensitivity.

2.7.

x̄ �1 + 3 + 7 + 15 + 25 + 39

6 � 15

σ2sv �(1 − 15)2 + (3 − 15)2 + (7 − 15)2

6 � 67 13

2.8. A normal distribution is symmetric. So the downside semi-variance and the upside semi-variance are equal,and the downside semi-variance is therefore half the total variance, or 50. The downside standard deviation is√

50 � 7.0711 .



2.9. This is a beta distribution. If you recognize it and are familiar with beta, you know that the mean is 2/3.Otherwise it is not hard to calculate:

E[X] �∫ 1

02x2dx �

23

The downside semi-variance is

σ2sv �

∫ 2/3

0

(x − 2

3

)2

2x dx

�

∫ 2/3

02x3 dx −

∫ 2/3

0

83 x2 dx +

∫ 2/3

0

89 x dx

�24

(23

)4

− 8/33

(23

)3

+8/92

(23

)2

� 0.032922

2.10. We need the first percentile of profits. Let it be x. Then

e−1000/x� 0.01

1000x

� − ln 0.01

x � − 1000ln 0.01 � 217.15

2.11. Let x be the VaR. Then (1000

x

)2

� 0.995

1000x

� 0.997497

x �1000

0.997497 � 1002.51

2.12. The 95th percentile of losses is the point with a 5% probability of losses above that point. Since the top intervalhas probability 3%, we need a subset of the (5000, 10000] interval with probability 2%. That interval has probability5%, so we need the top 2/5 of the interval, making 8000 the 95th percentile. The expected value of losses given thatthey’re above 8000 can be calculated using the double expectation formula:

TVaR(X) � E[X | X > 8000]� Pr(X ≤ 10000 | X > 8000)E[X | 8000 < X ≤ 10000] + Pr(X > 10000)E[X | X > 10000]

By uniformity, E[X ≤ 10000 | X > 8000] � 9000 and E[X | X > 10000] � 15000. So

TVaR(X) � 0.4(9000) + 0.6(15000) � 12,600

2.13. The 5th percentile of X is

e−x/1000� 0.95

x � −1000 ln 0.95 � 51.2933

The straightforward way to calculate the conditional expectation is to integrate x over the density function andthen divide by 0.05, the probability of X < 51.2933. The density function is 0.001e−x/1000.∫ 51.2933

00.001xe−x/1000 dx � −xe−x/1000

��51.2933

0+

∫ 51.2933

0e−x/1000 dx


QUIZ SOLUTIONS FOR LESSON 2 17

� −51.2933e−0.0512933 − 1000e−0.0512933+ 1000 � 1.27137

TVaR0.05(X) � 1.271370.05 � 25.4274

2.14. Average the top 5 numbers.

976 + 989 + 1005 + 1032 + 10505 � 1010.4

2.15. g(X) does not satisfy translation invariance since E[(X + c)2] , E[X]2 + c. It does not satisfy positivehomogeneity since E[(cX)2] , c E[X2]. Also, E[(X + Y)2] � E[X2] + 2 E[XY] + E[Y2]. Since E[XY] may be greaterthan 0, it does not satisfy subadditivity. However, it does satisfy monotonicity, since if X ≤ Y, and both X and Y arenonnegative, then X2 − Y2 ≤ 0 so E[X2] − E[Y2] ≤ 0.

2.16. g(X) does not satisfy translation invariance since E[√

X + c], E

[√X]+ c. It does not satisfy positive

homogeneity since E[√

cX], c E[√X]. For subadditivity, we need

E[√

X + Y]≤ E

[√X]+ E

[√Y]� E

[√X +√

Y]

This will be true if √x + y ≤ √x +√

y for all x , y ≥ 0. Square both sides of the inequality, and this is equivalent to√x y ≥ 0, which is true. So this risk measure is subadditive. It is also monotonic, since if X ≤ Y, then

√X − √Y ≤ 0

and therefore the expected value of√

X − √Y is less than 0.

Quiz Solutions

2-1. The present value of the free cash flows, in millions, is 1/(0.09 − 0.03) � 16.666667. The NPV is 16,666,667 −10,000,000 � 6,666,667 .2-2. Let n be the number of years the product sells. The NPV in millions is −9 + an .

−9 +1 − 1/1.1n

0.1 � 0

1 − 11.1n � 0.9

1.1n�

10.1 � 10

n ln 1.1 � ln 10n � 24.1589

Since we have no provision for fractional years, the answer is 25 , although this leads to a slightly positive NPV.2-3. The mean is 15. The sample semi-variance is

(5 − 15)2 + (10 − 15)25 � 25


Practice Exam 1

1. Options on XYZ stock trade on the Newark Exchange. Each option is for 100 shares. You are given that onMarch 21:

(i) 3000 options traded.(ii) The price of a share of stock was $40.(iii) The price of each option was $90.

Determine the total notional value of all of the options traded.

(A) 3,000 (B) 9,000 (C) 120,000 (D) 270,000 (E) 12,000,000

2. For American put options on a stock with identical expiry dates, you are given the following prices:

Strike price Put premium30 2.4035 6.40

For an American put option on the same stock with the same expiry date and strike price 38, which of thefollowing statements is correct?(A) The lowest possible price for the option is 8.80.(B) The highest possible price for the option is 8.80.(C) The lowest possible price for the option is 9.20.(D) The highest possible price for the option is 9.20.(E) The lowest possible price for the option is 9.40.

3. A company has 100 shares of ABC stock. The current price of ABC stock is 30. ABC stock pays no dividends.The company would like to guarantee its ability to sell the stock at the end of six months for at least 28.European call options on the same stock expiring in 6 months with exercise price 28 are available for 4.10.The continuously compounded risk-free interest rate is 5%.Determine the cost of the hedge.

(A) 73 (B) 85 (C) 99 (D) 126 (E) 141


497 Exam questions continue on the next page . . .

498 PRACTICE EXAMS

4. You are given the following prices for a stock:

Time PriceInitial 39After 1 month 39After 2 months 37After 3 months 43

Aportfolio of 3-monthAsian options, each based onmonthly averages of the stock price, consists of the following:

(i) 100 arithmetic average price call options, strike 36.(ii) 200 geometric average strike call options.(iii) 300 arithmetic average price put options, strike 41.

Determine the net payoff of the portfolio after 3 months.

(A) 1433 (B) 1449 (C) 1464 (D) 1500 (E) 1512

5. The price of a 6-month futures contract on widgets is 260.A 6-month European call option on the futures contract with strike price 256 is priced using Black’s formula.You are given:

(i) The continuously compounded risk-free rate is 0.04.(ii) The volatility of the futures contract is 0.25.

Determine the price of the option.

(A) 19.84 (B) 20.16 (C) 20.35 (D) 20.57 (E) 20.74

6. The beta of a stock is 0.8. The volatility of the stock is 0.3.The volatility of the market portfolio is 0.2.Calculate the non-diversifiable risk of the stock, as measured by volatility.

(A) 0.16 (B) 0.20 (C) 0.25 (D) 0.40 (E) 0.60

7. Investor A bought a 40-strike European call option expiring in 1 year on a stock for 5.50. Investor A earned aprofit of 6.44 at the end of the year.

Investor B bought a 45-strike European call option expiring in 1 year on the same stock at the same time, andearned a profit of 3.22 at the end of the year.

The continuously compounded risk-free interest rate is 2%.Determine the price of the 45-strike European call option.

(A) 3.67 (B) 3.71 (C) 3.75 (D) 3.78 (E) 3.82

8. Which of the following are inconsistent with the semi-strong form of the efficient market hypothesis but notwith the weak form?

I. Changes in a stock’s price in a week are positively correlated with changes in that stock’s price in the previousweek.

II. One can beat the market by using publicly available information.III. One can beat the market by using hard-to-get information.

(A) None (B) I only (C) II only (D) III only(E) The correct answer is not given by (A) , (B) , (C) , or (D) .


Exam questions continue on the next page . . .

PRACTICE EXAM 1 499

9. You own 100 shares of a stock whose current price is 42. You would like to hedge your downside exposureby buying 100 6-month European put options with a strike price of 40. You are given:

(i) The Black-Scholes framework is assumed.(ii) The continuously compounded risk-free interest rate is 5%.(iii) The stock pays no dividends.(iv) The stock’s volatility is 22%.

Determine the cost of the put options.

(A) 121 (B) 123 (C) 125 (D) 127 (E) 129

10. You are given the following information for a European call option expiring at the end of three years:

(i) The current price of the stock is 66.(ii) The strike price of the option is 70.(iii) The continuously compounded risk-free interest rate is 0.05.(iv) The continuously compounded dividend rate of the stock is 0.02.

The option is priced using a 1-period binomial tree with u � 1.3, d � 0.7.A replicating portfolio consists of shares of the underlying stock and a loan.Determine the amount borrowed in the replicating portfolio.

(A) 14.94 (B) 15.87 (C) 17.36 (D) 17.53 (E) 18.43

11. A company has a 25% probability of having 50 million in assets and a 75% probability of having 150 millionin assets at the end of one year. It has debt of 80 million due in one year. Bankruptcy costs are 20 million.

The cost of debt capital is 6% and the cost of equity capital is 18%.The corporate tax rate is 20%.Calculate the value of the company.

(A) 101.1 million (B) 108.1 million (C) 108.9 million (D) 112.8 million (E) 113.7 million

12. For European options on a stock having the same expiry and strike price, you are given:

(i) The stock price is 85.(ii) The strike price is 90.(iii) The continuously compounded risk free rate is 0.04.(iv) The continuously compounded dividend rate on the stock is 0.02.(v) A call option has premium 9.91.(vi) A put option has premium 12.63.

Determine the time to expiry for the options.

(A) 3 months (B) 6 months (C) 9 months (D) 12 months (E) 15 months



500 PRACTICE EXAMS

13. A portfolio of European options on a stock consists of a bull spread of calls with strike prices 48 and 60 and abear spread of puts with strike prices 48 and 60.

You are given:

(i) The options all expire in 1 year.(ii) The current price of the stock is 50.(iii) The stock pays dividends at a continuously compounded rate of 0.01.(iv) The continuously compounded risk-free interest rate is 0.05.

Calculate the price of the portfolio.

(A) 9.51 (B) 9.61 (C) 9.90 (D) 11.41 (E) 11.53

14. Stock ABC’s expected annual rate of return is 0.10 with volatility 0.25. Stock DEF’s expected annual rate ofreturn is 0.08 with volatility 0.30. An equally weighted portfolio of the two stocks has a volatility of 0.22.

Calculate the correlation between the two stock returns.

(A) 0.021 (B) 0.079 (C) 0.137 (D) 0.274 (E) 0.548

15. You are given the following graph of the profit on a position with derivatives:

20 30 40 50 60 70Stock Price

−20

−10

0

10

20

Profi

t

Determine which of the following positions has this profit graph.(A) Long forward(B) Short forward(C) Long collar(D) Long collared stock(E) Short collar

16. Which of the following behaviors may make the market portfolio inefficient?

I. Investors invest in stocks they are most familiar with.II. Investors seek sensation.III. Investors hang on to losers and sell winners.

(A) I and II only (B) I and III only (C) II and III only (D) I, II, and III(E) The correct answer is not given by (A) , (B) , (C) , or (D) .



PRACTICE EXAM 1 501

17. For a put option on a stock:

(i) The premium is 2.56.(ii) Delta is −0.62.(iii) Gamma is 0.09.(iv) Theta is −0.02 per day.

Calculate the delta-gamma-theta approximation for the put premium after 3 days if the stock price goes up by 2.

(A) 1.20 (B) 1.32 (C) 1.44 (D) 1.56 (E) 1.62

18. Which of the following are differences between a venture capital firm and a private equity firm?

I. Private equity firms deal with existing privately held firms while venture capital firms deal with start-upcompanies.

II. Private equity deals are larger.III. General partners of venture capital firms receive carried interest, unlike general partners of private equity firms.

(A) I and II (B) I and III (C) II and III (D) I, II, and III(E) The correct answer is not given by (A) , (B) , (C) , or (D) .

19. For an at-the-money European call option on a nondividend paying stock:

(i) The price of the stock follows the Black-Scholes framework(ii) The option expires at time t.(iii) The option’s delta is 0.5832.

Calculate delta for an at-the-money European call option on the stock expiring at time 2t.

(A) 0.62 (B) 0.66 (C) 0.70 (D) 0.74 (E) 0.82

20. You are given the following sample:

10 25 48 52 100 125

Calculate the sample downside standard deviation.

(A) 25.4 (B) 25.6 (C) 27.0 (D) 27.4 (E) 27.8

21. Gap options on a stock have six months to expiry, strike price 50, and trigger 49. You are given:

(i) The stock price is 45.(ii) The continuously compounded risk free rate is 0.08.(iii) The continuously compounded dividend rate of the stock is 0.02.

The premium for a gap call option is 1.68.Determine the premium for a gap put option.

(A) 4.20 (B) 5.17 (C) 6.02 (D) 6.96 (E) 7.95

22. Determine which of the following positions has the same cash flow as a short zero-coupon bond position.(A) Long stock and long forward(B) Long stock and short forward(C) Short stock and long forward(D) Short stock and short forward(E) Long forward and short forward



502 PRACTICE EXAMS

23. A 1-year American pound-denominated put option on euros allows the sale of e100 for £90. It is modeledwith a 2-period binomial tree based on forward prices. You are given

(i) The spot exchange rate is £0.8/e.(ii) The continuously compounded risk-free rate in pounds is 0.06.(iii) The continuously compounded risk-free rate in euros is 0.04.(iv) The volatility of the exchange rate of pounds to euros is 0.1.

Calculate the price of the put option.

(A) 8.92 (B) 9.36 (C) 9.42 (D) 9.70 (E) 10.00

24. You are conducting a break-even analysis on a project. The project has the following parameters:

(i) Initial investment: 16 million.(ii) Free cash flows in first year: 2 million.(iii) Rate of growth in cash flows: 3% per year.(iv) Cost of capital: 12% annual effective rate.(v) Project lifetime: infinite

Calculate the cost of capital to break even.

(A) 0.115 (B) 0.125 (C) 0.135 (D) 0.145 (E) 0.155

25. The price of an asset, X(t), follows the Black-Scholes framework. You are given that

(i) The continuously compounded expected rate of appreciation is 0.1.(ii) The volatility is 0.2.

Determine Pr(X(2)3 > X(0)3) .

(A) 0.63 (B) 0.65 (C) 0.67 (D) 0.69 (E) 0.71

26. A market-maker writes a 1-year call option and delta-hedges it. You are given:

(i) The stock’s current price is 100.(ii) The stock pays no dividends.(iii) The call option’s price is 4.00.(iv) The call delta is 0.76.(v) The call gamma is 0.08.(vi) The call theta is −0.02 per day.(vii) The continuously compounded risk-free interest rate is 0.05.

The stock’s price rises to 101 after 1 day.Estimate the market-maker’s profit.

(A) −0.04 (B) −0.03 (C) −0.02 (D) −0.01 (E) 0



PRACTICE EXAM 1 503

27. You are given the following decision tree for a project:

−1000

0

−1000

−800

−1500

−80050%

2000

50%

200

40% 1200

60%900

40% 2000

60%1000

50% 2000

50%1000

60% 2500

40%1200

70% 3000

30%1400

The interest rate is 0.Calculate the expected profit using the optimal strategy.

(A) 1000 (B) 1020 (C) 1040 (D) 1060 (E) 1080

28. You are given:

(i) The price of a stock is 40.(ii) The continuous dividend rate for the stock is 0.02.(iii) Stock volatility is 0.3.(iv) The continuously compounded risk-free interest rate is 0.06.

A 3-month at-the-money European call option on the stock is priced with a 1-period binomial tree. The tree isconstructed so that the risk-neutral probability of an up move is 0.5 and the ratio between the prices on the higherand lower nodes is e2σ

√h , where h is the amount of time between nodes in the tree.

Determine the resulting price of the option.

(A) 3.11 (B) 3.16 (C) 3.19 (D) 3.21 (E) 3.28

29. You are given the following information for two stocks:

Expected return VolatilityStock A 0.2 0.3Stock B 0.1 0.2

The correlation between the two stocks is −0.5.A portfolio consists of 16% Stock A and 84% Stock B.There is a more efficient portfolio of the two stocks having the same volatility.Determine the proportion of that portfolio invested in Stock A.

(A) 0.44 (B) 0.52 (C) 0.58 (D) 0.62 (E) 0.68



504 PRACTICE EXAMS

30. For a portfolio of call options on a stock:

Number of Call premiumshares of stock per share Delta

100 11.4719 0.6262100 11.5016 0.6517200 10.1147 0.9852

Calculate delta for the portfolio.

(A) 0.745 (B) 0.812 (C) 0.934 (D) 297.9 (E) 324.8

Solutions to the above questions begin on page 593.


Appendix A. Solutions for the Practice Exams

Answer Key for Practice Exam 11 E 11 C 21 B2 A 12 E 22 C3 E 13 D 23 E4 B 14 D 24 E5 A 15 C 25 E6 A 16 E 26 B7 C 17 C 27 C8 C 18 A 28 B9 E 19 A 29 C10 B 20 B 30 E

Practice Exam 1

1. [Lesson 1] Notional value is the value of the underlying asset. Here that is 100(40) � 4000 for each option,or 4000(3000) � 12,000,000 for all options. (E)

2. [Section 19.4] Options are convex, meaning that as the strike price increases, the rate of increase in the putpremium does not decrease. The rate of increase from 30 to 35 is (6.40− 2.40)/(35− 30) � 0.80, so the rate of increasefrom 35 to 38 must be at least (38 − 35)(0.80) � 2.40, making the price at least 6.40 + 2.40 � 8.80. Thus (A) is correct.

3. [Subsection 18.1] By put-call parity,

P � C + Ke−rt − Se−δt

� 4.10 + 28e−0.025 − 30 � 1.4087

For 100 shares, the cost is 100(1.4087) � 140.87 . (E)

4. [Section 27.1] The monthly arithmetic average of the prices is

39 + 37 + 433 � 39.6667

The monthly geometric average of the prices is

3√(39)(37)(43) � 39.5893

The payments on the options are:• The arithmetic average price call options with strike 36 pay 39.6667 − 36 � 3.6667.• The geometric average strike call options pay 43 − 39.5893 � 3.4107.• The arithmetic average price put options with strike 41 pay 41 − 39.6667 � 1.3333.

The total payment on the options is 100(3.6667) + 200(3.4107) + 300(1.3333) � 1448.8 . (B)


593

594 PRACTICE EXAM 1, SOLUTIONS TO QUESTIONS 5–11

5. [Section 24.3] By Black’s formula,

d1 �ln(260/256) + 0.5(0.252)(0.5)

0.25√

0.5� 0.17609

d2 � 0.17609 − 0.25√

0.5 � −0.00068N(d1) � N(0.17609) � 0.56989N(d2) � N(−0.00068) � 0.49973

C � 260e−0.02(0.56989) − 256e−0.02(0.49973) � 19.84 (A)

6. [Lesson 6] By formula (6.2), setting the portfolio equal to the market, beta is the quotient of the market riskof a stock over market volatility. So market risk of our stock, which is non-diversifiable, is 0.8(0.2) � 0.16 . (A)

7. [Section 16.1] LetS be thevalue of the stock at the endof oneyear. Profit for InvestorA isS−40−5.5e0.02 � 6.44.Therefore S � 52.05.

Let C be the call price for Investor B. For Investor B, profit was 52.05 − 45 − Ce0.02 � 3.22. Therefore, C � 3.75 .(C)

8. [Lesson 4] I is inconsistent even with the weak form. II is inconsistent with the semi-strong form but notwith the weak form. III in consistent even with the semi-strong form. (C)

9. [Lesson 24] For one share, Black-Scholes formula gives:

d1 �ln(42/40) + (

0.05 − 0 + 0.5(0.222))(0.5)0.22√

0.5� 0.55212

d2 � 0.55212 − 0.22√

0.5 � 0.39656N(−d2) � N(−0.39656) � 0.34585N(−d1) � N(−0.55212) � 0.29043

P � 40e−0.05(0.5)(0.34585) − 42(0.29043) � 1.2944

The cost of 100 puts is 100(1.2944) � 129.44 . (E)Note that this question has nothing to dowith delta hedging. The purchaser is merely interested in guaranteeing

that he receives at least 40 for each share, and does not wish to give up upside potential. A delta hedger gives upupside potential in return for keeping loss close to zero.

10. [Lesson 20] Cd � 0 and Cu � 1.3(66) − 70 � 15.8. By equation (20.2),

B � e−rt(uCd − dCu

u − d

)� e−0.15

(−0.7(15.8)0.6

)� −15.87

15.87 is borrowed. (B)

11. [Lesson 11] In millions, the value of debt capital is 0.25(50−20)+0.75(80) � 67.5 discounted at 0.06(1−0.2) �0.048. The value of equity capital is 0.25(0) + 0.75(70) � 52.5 discounted at 0.18. The company’s value is

67.51.048 +

52.51.18 � 108.900 (C)


PRACTICE EXAM 1, SOLUTIONS TO QUESTIONS 12–18 595

12. [Subsection 18.1] By put-call parity

12.63 − 9.91 � 90e−0.04t − 85e−0.02t

90e−0.04t − 85e−0.02t − 2.72 � 0

Let x � e−0.02t and solve the quadratic for x.

x �85 +

√852 + 4(90)(2.72)

2(90) �175.577

180 � 0.975428

The other solution to the quadratic leads to x < 0, which is impossible for x � e−0.02t . Now we solve for t.

e−0.02t� 0.975428

0.02t � − ln 0.975428 � 0.024879

t � 50(0.024879) � 1.244 (E)

13. [Section 17.3] This is a box spread. At expiry, the 48-strike call and put will require payment of 48, and the60-strike call and put will result in receiving 60, so the portfolio will pay 60 − 48 � 12. The present value of 12 is12e−0.05 � 11.41 . (D)

14. [Lesson 5] We will use formula (5.3). The average variance of the stocks is

12 (0.252

+ 0.32) � 00.07625

By formula (5.3),

0.222�

12 (0.1525) + 1

2 Cov(ABC,DEF)Solving for the covariance

Cov(ABC,DEF) � 2(0.0484 − 1

2 (0.07625))� 0.02055

The correlation is 0.02055/(0.25 · 0.3) � 0.274 . (D)

15. [Section 17.3] A long collar has this graph. (C) A short forward wouldn’t have the flat section. Longforwards and short collars increase in value with increasing stock prices. A collared stock has flat lines on the leftand right.

16. [Section 8.1] I and II are idiosyncratic, so they do not make the market portfolio inefficient. III is systematic,so it makes the market portfolio inefficient.(E)

17. [Section 26.2] Theta is expressed per day of decrease, so we just have to multiply it as given by 3. Thus thechange in price is

∆ε + 0.5Γε2+ θh � −0.62(2) + 0.5(0.09)(22) − 0.02(3) � −1.12

The new price is 2.56 − 1.12 � 1.44 . (C)

18. [Section 12.1] I and II are true. Fees for private equity firms are similar to fees for venture capital firms. (A)



19. [Section 25.1] Delta is e−δt N(d1), or N(d1) for a nondividend paying stock. Since the option is at-the-money,

d1 �(r + 0.5σ2)t

σ√

t�

r + 0.5σ2

σ

√t

So doubling time multiplies d1 by√

2.

N(d1) � 0.5832d1 � N−1(0.5832) � 0.2101

d1√

2 � (0.2101)(1.4142) � 0.2971

N(0.2971) � 0.6168 (A)

20. [Section 2.3]

x̄ � 60√(10 − 60)2 + (25 − 60)2 + (48 − 60)2 + (52 − 60)2

6 � 25.6027 (B)

21. [Section 28.1] For gap options, put-call parity applies with the strike price. If you buy a call and sell a put,if ST > K2 (the trigger price) you collect St and pay K1, and if St < K2 you pay K1 and collect St which is the sameas collecting St and paying K1, so

C − P � Se−δt − K1e−rt

In this problem,P � C + K1e−rt − Se−δt

� 1.68 + 50e−0.04 − 45e−0.01� 5.167 (B)

22. [Section 14.4] A synthetic forward is a long stock plus a short bond. So a short bond is a long forward plusa short stock. (C)

23. [Section 21.3] The 6-month forward rate of euros in pounds is e(0.06−0.04)(0.5) � e0.01 � 1.01005. Up and downmovements, and the risk-neutral probability of an up movement, are

u � e0.01+0.1√

0.5� 1.08406

d � e0.01−0.1√

0.5� 0.94110

p∗ �1.01005 − 0.941101.08406 − 0.94110 � 0.4823

1 − p∗ � 1 − 0.4823 � 0.5177

The binomial tree is shown in Figure A.1. At the upper node of the second column, the put value is calculated as

Pu � e−0.03(0.5177)(0.08384) � 0.04212

At the lower node of the second column, the put value is calculated as

Ptentatived � e−0.03 ((0.4823)(0.08384) + (0.5177)(0.19147)) � 0.13543

but the exercise value 0.9− 0.75288 � 0.14712 is higher so it is optimal to exercise. At the initial node, the calculatedvalue of the option is

Ptentative� e−0.03 ((0.4823)(0.04212) + (0.5177)(0.14712)) � 0.09363

Since 0.9− 0.8 � 0.1 > 0.09363, it is optimal to exercise the option immediately, so its value is 0.10 (which means thatsuch an option would never exist), and the price of an option for e100 is 100(0.10) � 10 . (E)


PRACTICE EXAM 1, SOLUTIONS TO QUESTIONS 24–27 597

0.867250.04212

0.80.1

0.752880.14712

0.940140

0.816160.08384

0.708530.19147

Figure A.1: Exchange rates and option values for put option of question 23

24. [Section 2.2] Let r be the cost of capital. The NPV in millions is

NPV � −16 +2

r − 0.03 � 0

1r − 0.03 � 8

8r − 0.24 � 1

r �1.24

8 � 0.155 (E)

25. [Section 23.2] The fraction X(2)/X(0) follows a lognormal distribution with parameters m � 2(0.1 −

0.5(0.22)) � 0.16 and v � 0.2√

2. Cubing does not affect inequalities, so the requested probability is the sameas Pr

(ln X(2) − ln X(0) > 0

), which is

1 − N(−0.160.2√

2

)� N(0.56569) � 0.7142 (E)

26. [Section 26.2] By formula (26.3) with ε � 1 and h � 1/365,

Market-Maker Profit � −0.5Γε2 − θh − rh(S∆ − C(S))

� −0.5(0.08)(12) + 0.02 − 0.05365 [(100)(0.76) − 4]

� −0.04 + 0.02 − 0.00986 � −0.02986 (B)

27. [Lesson 30] For the bottom two nodes, expected profit is 0.4(1200) + 0.6(900) � 1020. The expected profitsat the four information nodes in the fourth column, one period before the end, are, in order from top to bottom:

0.7(3000) + 0.3(1400) � 25200.6(2500) + 0.4(1200) � 19800.5(2000) + 0.5(1000) � 15000.4(2000) + 0.6(1000) � 1400



Pulling back to the third column, these become

2520 − 1500 � 10201980 − 800 � 11801500 − 800 � 7001400 − 1000 � 400

At the upper decision node in the second column, 1180 is selected. At the lower decision node in the second column,700 is selected. At the information node in the first column, expected profit is 0.5(1180+2000)+0.5(700+200) � 2040.After subtracting 1000, this yields 1040 at the initial decision node, which is greater than 1020, so it is the optimalprofit. (C)

28. [Lesson 20] The risk-neutral probability is

0.5 � p∗ �e(r−δ)h − d

u − d�

e(0.06−0.02)(0.25) − du − d

�e0.01 − d

u − d

but u � de2σ√

h � de2(0.3)(1/2) � de0.3, so

e0.01 − d � 0.5(e0.3d − d

)e0.01

� d(0.5(e0.3 − 1) + 1

)� 1.17493d

d �e0.01

1.17493 � 0.85967

u � 0.85967e0.3� 1.16043

The option only pays at the upper node. The price of the option is

C � e−rh p∗(Su − K) � e−0.06(0.25)(0.5)(40(1.16043) − 40)� 3.1609 (B)

29. [Lesson 5] The variance of the portfolio is

0.162(0.32) + 0.842(0.22) + 2(−0.5)(0.16)(0.84)(0.2)(0.3) � 0.022464

Let p be the proportion invested in Stock A in the more efficient portfolio. Then

0.09p2+ 0.04(1 − p)2 − 0.06p(1 − p) � 0.022464

0.19p2 − 0.14p + 0.017536 � 0

p �0.14 +

√0.006273

0.38 � 0.5768 (C)

30. [Subsection 25.1.7] Delta for a portfolio of options on a single stock is the sum of the individual deltas ofthe options.

100(0.6262) + 100(0.6517) + 200(0.9852) � 324.8 (E)


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